254 |
254 |
255 \nn{Almost certainly we need a little more than the above axiom. |
255 \nn{Almost certainly we need a little more than the above axiom. |
256 More specifically, in order to bootstrap our way from the top dimension |
256 More specifically, in order to bootstrap our way from the top dimension |
257 properties of identity morphisms to low dimensions, we need regular products, |
257 properties of identity morphisms to low dimensions, we need regular products, |
258 pinched products and even half-pinched products. |
258 pinched products and even half-pinched products. |
259 I'm not sure what the best way to cleanly axiomatize the properties of these various is. |
259 I'm not sure what the best way to cleanly axiomatize the properties of these various |
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260 products is. |
260 For the moment, I'll assume that all flavors of the product are at |
261 For the moment, I'll assume that all flavors of the product are at |
261 our disposal, and I'll plan on revising the axioms later.} |
262 our disposal, and I'll plan on revising the axioms later.} |
262 |
263 |
263 \nn{current idea for fixing this: make the above axiom a ``preliminary version" |
264 \nn{current idea for fixing this: make the above axiom a ``preliminary version" |
264 (as we have already done with some of the other axioms), then state the official |
265 (as we have already done with some of the other axioms), then state the official |
299 We define a map |
300 We define a map |
300 \begin{eqnarray*} |
301 \begin{eqnarray*} |
301 \psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
302 \psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
302 a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
303 a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
303 \end{eqnarray*} |
304 \end{eqnarray*} |
304 \nn{need to say something somewhere about pinched boundary convention for products} |
305 (See Figure \ref{glue-collar}.) |
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306 \begin{figure}[!ht]\begin{equation*} |
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307 \mathfig{.9}{tempkw/glue-collar} |
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308 \end{equation*}\caption{Extended homeomorphism.}\label{glue-collar}\end{figure} |
305 We will call $\psi_{Y,J}$ an extended isotopy. |
309 We will call $\psi_{Y,J}$ an extended isotopy. |
306 \nn{or extended homeomorphism? see below.} |
310 \nn{or extended homeomorphism? see below.} |
307 \nn{maybe remark that in some examples (e.g.\ ones based on sub cell complexes) |
311 \nn{maybe remark that in some examples (e.g.\ ones based on sub cell complexes) |
308 extended isotopies are also plain isotopies, so |
312 extended isotopies are also plain isotopies, so |
309 no extension necessary} |
313 no extension necessary} |
358 |
362 |
359 \medskip |
363 \medskip |
360 |
364 |
361 The alert reader will have already noticed that our definition of (plain) $n$-category |
365 The alert reader will have already noticed that our definition of (plain) $n$-category |
362 is extremely similar to our definition of topological fields. |
366 is extremely similar to our definition of topological fields. |
363 The only difference is that for the $n$-category definition we restrict our attention to balls |
367 The main difference is that for the $n$-category definition we restrict our attention to balls |
364 (and their boundaries), while for fields we consider all manifolds. |
368 (and their boundaries), while for fields we consider all manifolds. |
365 \nn{also: difference at the top dimension; fix this} |
369 (A minor difference is that in the category definition we directly impose isotopy |
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370 invariance in dimension $n$, while in the fields definition we have non-isotopy-invariant fields |
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371 but then mod out by local relations which imply isotopy invariance.) |
366 Thus a system of fields determines an $n$-category simply by restricting our attention to |
372 Thus a system of fields determines an $n$-category simply by restricting our attention to |
367 balls. |
373 balls. |
368 The $n$-category can be thought of as the local part of the fields. |
374 This $n$-category can be thought of as the local part of the fields. |
369 Conversely, given an $n$-category we can construct a system of fields via |
375 Conversely, given an $n$-category we can construct a system of fields via |
370 \nn{gluing, or a universal construction} |
376 a colimit construction; see below. |
371 \nn{see subsection below} |
377 |
372 |
378 %\nn{Next, say something about $A_\infty$ $n$-categories and ``homological" systems |
373 \nn{Next, say something about $A_\infty$ $n$-categories and ``homological" systems |
379 %of fields. |
374 of fields. |
380 %The universal (colimit) construction becomes our generalized definition of blob homology. |
375 The universal (colimit) construction becomes our generalized definition of blob homology. |
381 %Need to explain how it relates to the old definition.} |
376 Need to explain how it relates to the old definition.} |
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377 |
382 |
378 \medskip |
383 \medskip |
379 |
384 |
380 \nn{these examples need to be fleshed out a bit more} |
385 \nn{these examples need to be fleshed out a bit more} |
381 |
386 |
403 \nn{need to say something about fundamental classes, or choose $\alpha$ carefully} |
408 \nn{need to say something about fundamental classes, or choose $\alpha$ carefully} |
404 |
409 |
405 \item Given a traditional $n$-category $C$ (with strong duality etc.), |
410 \item Given a traditional $n$-category $C$ (with strong duality etc.), |
406 define $\cC(X)$ (with $\dim(X) < n$) |
411 define $\cC(X)$ (with $\dim(X) < n$) |
407 to be the set of all $C$-labeled sub cell complexes of $X$. |
412 to be the set of all $C$-labeled sub cell complexes of $X$. |
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413 (See Subsection \ref{sec:fields}.) |
408 For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear |
414 For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear |
409 combinations of $C$-labeled sub cell complexes of $X$ |
415 combinations of $C$-labeled sub cell complexes of $X$ |
410 modulo the kernel of the evaluation map. |
416 modulo the kernel of the evaluation map. |
411 Define a product morphism $a\times D$ to be the product of the cell complex of $a$ with $D$, |
417 Define a product morphism $a\times D$ to be the product of the cell complex of $a$ with $D$, |
412 and with the same labeling as $a$. |
418 and with the same labeling as $a$. |
418 \nn{refer elsewhere for details?} |
424 \nn{refer elsewhere for details?} |
419 |
425 |
420 \item Variation on the above examples: |
426 \item Variation on the above examples: |
421 We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
427 We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
422 for example product boundary conditions or take the union over all boundary conditions. |
428 for example product boundary conditions or take the union over all boundary conditions. |
423 \nn{maybe should not emphasize this case, since it's ``better" in some sense |
429 %\nn{maybe should not emphasize this case, since it's ``better" in some sense |
424 to think of these guys as affording a representation |
430 %to think of these guys as affording a representation |
425 of the $n{+}1$-category associated to $\bd F$.} |
431 %of the $n{+}1$-category associated to $\bd F$.} |
426 |
432 |
427 \item \nn{should add bordism $n$-cat} |
433 \item Here's our version of the bordism $n$-category. |
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434 For a $k$-ball $X$, $k<n$, define $\cC(X)$ to be the set of all $k$-dimensional |
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435 submanifolds $W$ of $X\times \r^\infty$ such that the projection $W \to X$ is transverse |
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436 to $\bd X$. |
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437 For $k=n$ define $\cC(X)$ to be homeomorphism classes (rel boundary) of such submanifolds; |
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438 we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
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439 $W\to W'$ which restricts to the identity on the boundary. |
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440 |
428 |
441 |
429 \end{itemize} |
442 \end{itemize} |
430 |
443 |
431 |
444 |
432 Examples of $A_\infty$ $n$-categories: |
445 Examples of $A_\infty$ $n$-categories: |
461 (If $k = n$ and our $k$-categories are enriched, then |
474 (If $k = n$ and our $k$-categories are enriched, then |
462 $\cC(W)$ will have additional structure; see below.) |
475 $\cC(W)$ will have additional structure; see below.) |
463 $\cC(W)$ will be the colimit of a functor defined on a category $\cJ(W)$, |
476 $\cC(W)$ will be the colimit of a functor defined on a category $\cJ(W)$, |
464 which we define next. |
477 which we define next. |
465 |
478 |
466 Define a permissible decomposition of $W$ to be a decomposition |
479 Define a permissible decomposition of $W$ to be a cell decomposition |
467 \[ |
480 \[ |
468 W = \bigcup_a X_a , |
481 W = \bigcup_a X_a , |
469 \] |
482 \] |
470 where each $X_a$ is a $k$-ball. |
483 where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. |
471 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
484 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
472 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
485 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
473 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
486 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
474 (The objects of $\cJ(W)$ are permissible decompositions of $W$, and there is a unique |
487 (The objects of $\cJ(W)$ are permissible decompositions of $W$, and there is a unique |
475 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
488 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
499 (If $\dim(W) = n$ then we need to also make use of the monoidal |
512 (If $\dim(W) = n$ then we need to also make use of the monoidal |
500 product in the enriching category. |
513 product in the enriching category. |
501 \nn{should probably be more explicit here}) |
514 \nn{should probably be more explicit here}) |
502 |
515 |
503 Finally, define $\cC(W)$ to be the colimit of $\psi_\cC$. |
516 Finally, define $\cC(W)$ to be the colimit of $\psi_\cC$. |
504 In the plain (non-$A_\infty$) case, this means that |
517 When $k<n$ or $k=n$ and we are in the plain (non-$A_\infty$) case, this means that |
505 for each decomposition $x$ there is a map |
518 for each decomposition $x$ there is a map |
506 $\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps |
519 $\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps |
507 above, and $\cC(W)$ is universal with respect to these properties. |
520 above, and $\cC(W)$ is universal with respect to these properties. |
508 In the $A_\infty$ case, it means |
521 When $k=n$ and we are in the $A_\infty$ case, it means |
509 \nn{.... need to check if there is a def in the literature before writing this down; |
522 homotopy colimit. |
510 homotopy colimit I think} |
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511 |
523 |
512 More concretely, in the plain case enriched over vector spaces, and with $\dim(W) = n$, we can take |
524 More concretely, in the plain case enriched over vector spaces, and with $\dim(W) = n$, we can take |
513 \[ |
525 \[ |
514 \cC(W) = \left( \oplus_x \psi_\cC(x)\right) \big/ K |
526 \cC(W) = \left( \oplus_x \psi_\cC(x)\right) \big/ K |
515 \] |
527 \] |
517 $a\in \psi_\cC(x)$ for some decomposition $x$, $x\le y$, and $g: \psi_\cC(x) |
529 $a\in \psi_\cC(x)$ for some decomposition $x$, $x\le y$, and $g: \psi_\cC(x) |
518 \to \psi_\cC(y)$ is value of $\psi_\cC$ on the antirefinement $x\to y$. |
530 \to \psi_\cC(y)$ is value of $\psi_\cC$ on the antirefinement $x\to y$. |
519 |
531 |
520 In the $A_\infty$ case enriched over chain complexes, the concrete description of the colimit |
532 In the $A_\infty$ case enriched over chain complexes, the concrete description of the colimit |
521 is as follows. |
533 is as follows. |
522 \nn{should probably rewrite this to be compatible with some standard reference} |
534 %\nn{should probably rewrite this to be compatible with some standard reference} |
523 Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions. |
535 Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions. |
524 Such sequences (for all $m$) form a simplicial set. |
536 Such sequences (for all $m$) form a simplicial set. |
525 Let |
537 Let |
526 \[ |
538 \[ |
527 V = \bigoplus_{(x_i)} \psi_\cC(x_0) , |
539 V = \bigoplus_{(x_i)} \psi_\cC(x_0) , |
829 W = (\bigcup_a X_a) \cup (\bigcup_{i,b} M_{ib}) , |
842 W = (\bigcup_a X_a) \cup (\bigcup_{i,b} M_{ib}) , |
830 \] |
843 \] |
831 where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and |
844 where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and |
832 each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$, |
845 each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$, |
833 with $M_{ib}\cap\bd_i W$ being the marking. |
846 with $M_{ib}\cap\bd_i W$ being the marking. |
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847 \nn{need figure} |
834 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
848 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
835 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
849 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$. |
836 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
850 This defines a partial ordering $\cJ(W)$, which we will think of as a category. |
837 (The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique |
851 (The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique |
838 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |
852 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.) |